CWM book,ends,category theory,natural transformation

Question

I have a problem in the MacLane's book Categories for the working mathematician. On page 223,the chapter on Ends,he has two functors $U,V:C\rightarrow X$ and defines a dinatural transformation
$\tau:Y \xrightarrow{\cdot\cdot} \hom(U-,V-)$, for a fixed $Y \in Set$,with components
$\tau_c:Y\rightarrow \hom_X(Uc,Vc)$ that assigns to each $c \in C$ an arrow
$\tau_{c,y}:Uc\rightarrow Vc$ in $X$, such that for every arrow $f:b\rightarrow c$ in $C$ one has the wedge condition $Vf \circ \tau_{b,y}=\tau_{c,y} \circ Uf$ .Hence I understand that $\tau_{\_,y}$ will be a natural transformation $U\xrightarrow{\cdot} V$ for each fixed $y$. What I do not understand is why this wedge condition is required: why it holds,what is it implied by.We have dinatural transformation
$\tau:Y\xrightarrow{\cdot\cdot} \hom_X(U-,V-)$,and the functors $U$ is dummy in its second variable and $V$ is dummy in its first variable as functors $C^{op}\times C \rightarrow X$.

Answer 1

I tried to answer your other similar question and then I noticed this.

I think you misunderstood something: of course you can think of a functor $C\to X$ as a mute bifunctor $C\times C^{op}\to X$, but that's not the point of the exercise, there's no dinatural transformation you have to find between $U$ and $V$... the claim simply says that when you take two functors $U,V\colon C\to X$ and then you compose $\hom\circ(U\times V)$, then you get a functor $\hom(U, V)\colon C^{op}\times C\to Set$ whose end is the set of natural transformations between $U$ and $V$.

When $U=V=1$, you get a pleasant and intrinsic invariant of your category, the "Hochschild homology", see here.

I'd like to advise you to have a look at this note I'm writing in spare times.